Prediction after a Horizon of Predictability: Nonpredictable Points and Partial Multistep Prediction for Chaotic Time Series

Abstract

This paper introduces several novel strategies for multi-step-ahead prediction of chaotic time series. Introducing a concept of “generalized z-vectors” (vectors of nonsuccessive time series observations) makes it possible to generate sets of possible prediction values for each point we are trying to predict. Through examining these sets, unified predictions are calculated, which are in turn used as a basis for predicting subsequent points. The key difference between the strategy presented in this paper and its conventional counterparts is the concept of “nonpredictable” points (points which the algorithm categorized as “incalculable” and excluded from the calculations altogether). The results obtained for the benchmark and real-world time series indicate that while typically the number of nonpredictable points tends to grow exponentially with the number of steps ahead to be predicted, the average error for predicted points remains small and nearly constant. Thus, we redefine the problem of multi-step-ahead prediction as a two-objective optimization problem: on one hand, we aim to minimize the number of nonpredictable points and the average error among the predictable ones. The resulting strategy demonstrates accurate results for both benchmark and real-world time series, with the number of predicted steps exceeding that of any other published algorithm.